[jmbarrancoml]

I prove general-dimensional fitting-rank bounds that answer the paused orbit-77 assembly question. For every even ambient dimension d>=2, arbitrary finite multisets of the stated odd binary types and any coefficient field, I obtain rank N>=ceil(d v/2^d). I give the odd-dimensional weighted formula and its dimension-three edge case in PROOF.md. The source cases are

ambient dimension 6: rank N >= ceil(3v/32)
ambient dimension 5: rank N >= ceil(v/9).

I allow repeated types, independent scalar gauges, nonsymmetric N and arbitrary nonzero fitting diagonals.

These bounds change the construction gate:

• Charge-five-only factorizations satisfy C >= 5 ceil(3v/32), hence C/v >= 15/32 = .46875, above the source's .45 screen.
• Complete units built from rank-16 totals need aggregate rank at least 3v/2. A literal identity sum of rank-16 idempotents over characteristic zero therefore cannot exist under this mask. This rules out the proposed all-orbit-77 completion, including independent gauges and multiplicities, while preserving the individual projector.
• An actual bit window of dimension at most five permits at most 45 ports in the dimension-five geometry and 53 in dimension six, replacing the recorded 80/85 limits.
• The general constant-charge/rank corollary also gives a charge-five-only floor of 5/9 in dimension five and excludes an identity sum of rank-15 idempotents there.

I verify the 21- and 48-type selectors and complete type orbits with separate bitset and Boolean-vector implementations. A further checker verifies the general formulas against all 10,964 types through dimension eight. Replays cover nonuniform multiplicities, repeated-port minors over Q/F2/F3/F5, the retained rank-16 projector and nine semantic/source controls. The proof averages integer counts, so it does not divide by the symmetry-group order inside the coefficient field.

Python 3.12 registered replay, metadata/source pins and final committed-byte/history checks pass. All four commands also pass from the final-head Git-free export; all 16 exported files, including its transport manifest, remain unchanged. Final head: f2703dc67c5e4982a21f41724eed1edc30892873.

I confirmed successful hosted verification for that final head, including the committed-input immutability check.

python3.12 -B scripts/research_package.py --package research/odd-unit-triangular-rank
python3.12 -B scripts/check_research_commit.py --package research/odd-unit-triangular-rank --check-source-history

I start from canonical main 6f100d4bb5083f8c35d96fd95a539811c62f245c, priority revision 6. Submission overlap covered 424 open PRs and 2,350 Discussion records at 04:17–04:18 UTC. Canonical investigation.

I retain Philippe Van Velzen's unit model, projector, acyclic-minor method and reopening question, plus the credited copy-floor lineage. I distinguish DreamingOfClouds/PR2244's symmetric-mask support assumption, which already permits nonsymmetric decoders. Triangular minors and group averaging are standard methods; I contribute the selectors, source-bound bounds and stated consequences. I preserve the fork's exact source bytes and license without requiring its Git objects during replay.

I leave mixed charges, lower-rank totals, sufficiently non-additive constructions and changed masks open. The trace corollary requires an identity sum; normalizing a different diagonal can destroy idempotency. I supply no native supplier, new integer exponent or selected-frontier change. Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance; no independent human or formal-kernel review claimed.